Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.The general solution to a linear equation can be written as y = yc + yp. P 1 ( x ) Q 1 ( y ) + P 2 ( x ) Q 2 ( y ) d y d x = 0 P 1 ( x ) Q 1 ( y ) d x + P 2 ( x ) Q 2 ( y ) d y = 0 {displaystyle {egin{aligned}P_{1}(x)Q_{1}(y)+P_{2}(x)Q_{2}(y),{frac {dy}{dx}}&=0\P_{1}(x)Q_{1}(y),dx+P_{2}(x)Q_{2}(y),dy&=0end{aligned}}} d y d x = F ( x ) d y = F ( x ) d x {displaystyle {egin{aligned}{frac {dy}{dx}}&=F(x)\dy&=F(x),dxend{aligned}}} d y d x = F ( y ) d y = F ( y ) d x {displaystyle {egin{aligned}{frac {dy}{dx}}&=F(y)\dy&=F(y),dxend{aligned}}} P ( y ) d y d x + Q ( x ) = 0 P ( y ) d y + Q ( x ) d x = 0 {displaystyle {egin{aligned}P(y){frac {dy}{dx}}+Q(x)&=0\P(y),dy+Q(x),dx&=0end{aligned}}} d y d x = F ( y x ) {displaystyle {frac {dy}{dx}}=Fleft({frac {y}{x}} ight),!} y M ( x y ) + x N ( x y ) d y d x = 0 y M ( x y ) d x + x N ( x y ) d y = 0 {displaystyle {egin{aligned}yM(xy)+xN(xy),{frac {dy}{dx}}&=0\yM(xy),dx+xN(xy),dy&=0end{aligned}}} ln ⁡ ( C x ) = ∫ x y N ( λ ) d λ λ [ N ( λ ) − M ( λ ) ] {displaystyle ln(Cx)=int ^{xy}{frac {N(lambda ),dlambda }{lambda }},!} M ( x , y ) d y d x + N ( x , y ) = 0 M ( x , y ) d y + N ( x , y ) d x = 0 {displaystyle {egin{aligned}M(x,y){frac {dy}{dx}}+N(x,y)&=0\M(x,y),dy+N(x,y),dx&=0end{aligned}}} where Y(y) and X(x) are functions from the integrals rather than constant values, which are set to make the final function F(x, y) satisfy the initial equation. M ( x , y ) d y d x + N ( x , y ) = 0 M ( x , y ) d y + N ( x , y ) d x = 0 {displaystyle {egin{aligned}M(x,y){frac {dy}{dx}}+N(x,y)&=0\M(x,y),dy+N(x,y),dx&=0end{aligned}}} ∂ ( μ M ) ∂ x = ∂ ( μ N ) ∂ y {displaystyle {frac {partial (mu M)}{partial x}}={frac {partial (mu N)}{partial y}},!} F ( x , y ) = ∫ y μ ( x , λ ) M ( x , λ ) d λ + ∫ x μ ( λ , y ) N ( λ , y ) d λ + Y ( y ) + X ( x ) = C {displaystyle {egin{aligned}F(x,y)=&int ^{y}mu (x,lambda )M(x,lambda ),dlambda +int ^{x}mu (lambda ,y)N(lambda ,y),dlambda \&+Y(y)+X(x)=Cend{aligned}}} d 2 y d x 2 = F ( y ) {displaystyle {frac {d^{2}y}{dx^{2}}}=F(y),!} d y d x + P ( x ) y = Q ( x ) {displaystyle {frac {dy}{dx}}+P(x)y=Q(x),!} d 2 y d x 2 + b d y d x + c y = r ( x ) {displaystyle {frac {d^{2}y}{dx^{2}}}+b{frac {dy}{dx}}+cy=r(x),!} Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.If b2 > 4c, then ∑ j = 0 n b j d j y d x j = r ( x ) {displaystyle sum _{j=0}^{n}b_{j}{frac {d^{j}y}{dx^{j}}}=r(x),!} Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.Since αj are the solutions of the polynomial of degree n: ∏ j = 1 n ( α − α j ) = 0 {displaystyle prod _{j=1}^{n}(alpha -alpha _{j})=0,!} , then: In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a 0 ( x ) {displaystyle a_{0}(x)} , ..., a n ( x ) {displaystyle a_{n}(x)} and b ( x ) {displaystyle b(x)} are arbitrary differentiable functions that do not need to be linear, and y ′ , … , y ( n ) {displaystyle y',ldots ,y^{(n)}} are the successive derivatives of the unknown function y of the variable x. Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example Riccati equation). Some ODEs can be solved explicitly in terms of known functions and integrals. When that is not possible, the equation for computing the Taylor series of the solutions may be useful. For applied problems, numerical methods for ordinary differential equations can supply an approximation of the solution. Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations. Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modelling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes). Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert, and Euler. A simple example is Newton's second law of motion — the relationship between the displacement x and the time t of an object under the force F, is given by the differential equation